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Beschreibung:
The reconstruction and analysis of sparse signals is a common and widely studied problem in signal processing, for example in wireless telecommunication or power system theory. Hereby, most recovery methods exploit structures or special properties of the functions which are to be reconstructed. Particularly interesting are methods which aim to recover functions which possess a sparse representation in a given basis and use only a small set of sampling values. One of the most widely used methods is the so called Prony method, which is a deterministic method for the recovery of sparse exponential expansions. In recent year a generalization of Prony’s method for the reconstruction of spars expansion of eigenfunctions of certain linear operators has been introduced by Peter&Plonka in 2013. While some examples of suitable linear operators were given by Peter&Plonka, e.g., the shift operator as well as certain differential operators, the sample values needed for the reconstruction are not always accessible in practice. This leads to the following question. Can we find other suitable linear operators with meaningful structured functions as eigenfunctions and easily accessible sample values? Based on this question we investigate which structured functions can be recovered using only easily accessible sample values. Using the theory of one-parameter semigroups we derive a framework of so called generalized shift operators and their eigenfunctions, so-called generalized exponential sums, which covers all previously given examples for the generalized Prony method. Furthermore, we elaborate on the connection between generalized shift operators and linear differential operators and present a Prony based algorithm for the reconstruction of sparse generalized exponential expansion. Additionally, we present a new Prony based algorithm for the reconstruction of sparse expansions into orthogonal polynomials of length M using generating functions. Finally, we also consider the numerical analysis of the Prony method for ...